Swing is the "new" AWT. Sun decided they wanted it to be more versatile, so they made a lot new components (practically all of them are new I think).
A JFrame is a Frame with more stuff. :D
A JPanel is a Panel with more stuff. :D

Also, they added the content pane to the JFrame. I believe you used to be able to add things directly... no more.
Also note that javax.swing has some new stuff, like JOptionPane. All in all, think of Swing as an update to awt.

SWT is the software component that delivers native widget functionality for the Eclipse platform in an operating system independent manner. It is analogous to AWT/Swing in Java with a difference - SWT uses a rich set of native widgets. Even in an ideal situation, industrial strength cross platform widget libraries are very difficult to write and maintain. This is due to the inherent complexity of widget systems and the many subtle differences between platforms. There are several basic approaches that have helped significantly to reduce the complexity of the problem and deliver high quality libraries. This article discusses one of them, the low level implementation techniques used to implement SWT on different platforms. Examples are drawn from the Windows® and Motif implementations.